Sub-Riemannian structures on 3D lie groups

By Andrei A. Agrachev, Davide Barilari. 2010

151 citations recorded by the source corpus.

We give a complete classification of left-invariant sub-Riemannian structures on three-dimensional Lie groups in terms of the basic differential invariants. As a consequence, we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups SL(2) and A+($ \mathbb{R} $) × S1, where A+($ \mathbb{R} $) denotes the group of orientation preserving affine maps on the real line.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/244aef597f4771251ce9001beba69513b28b284b

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries (2018, unknown): “There are three-dimensional examples, see [1].”
A Comprehensive Introduction to Sub-Riemannian Geometry (2019, methodology): “The explicit expression given here is taken from [AB12] (cf.”
Small time heat kernel asymptotics at the sub-Riemannian cut locus (2012, background): “The heat kernel on sub-Riemannian manifolds has been an object of attention starting from the late 70s [13, 15, 16, 18, 20, 25, 31, 37, 40, 53, 54], as have the geodesics and cut and conjugate loci of such manifolds [2, 3, 5, 8, 9, 24, 28, 42, 51].”
Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry (2011, methodology): “For a geometric interpretation of these invariants and an intrinsic definition one can see [5, 1].”